My brother heard from someone that she heard that women are exempt from saying Tachanun and therefore discouraged from saying it. Is this so? If so, why? Do you know sources for this?

UPDATE: I searched for tachanun women and discovered this article as the first search result, which says that women are exempt but doesn't say anything about their being discouraged. The sources cited (all [near-]contemporary, if I'm not mistaken) apparently offer "various reasons." I don't have those books, so I don't know what the reasons are. Perhaps someone here does.

1 Answer
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Of the three books mentioned in the note in that article, Machazeh Eliyahu is available online. The page where he discusses this is here. Essentially, he says, it boils down to these two considerations:

One reason for saying Tachanun in the first place is so that we should pray in three different bodily positions - sitting, standing, and with head down. Since women are anyway not obligated in many other parts of prayer (specifically, most of the sitting parts), there is no particular reason for them to recite Tachanun with head down. (Although it occurs to me that this works only according to the Ashkenazic rite, where Tachanun consists essentially just of the prayer recited while "falling on the face." In the Sephardic and Chassidic versions, there are other parts of Tachanun; this reason wouldn't explain why women don't say these.)

In general, Tachanun is less obligatory than the other parts of the prayers, which indeed is why there are many circumstances and dates when it is omitted. So we may assume that Jewish women in general never accepted it as a fixed obligation (much like with Maariv, which is also technically a voluntary prayer).